osp(1|2)-trivial deformation of osp(2|2)-modules structure on the spaces of symbols Sd2 of differential operators acting on the space of weighted densities Fd2

Let osp(2|2) be the orthosymplectic Lie superalgebra and osp(1|2) a Lie subalgebra of osp(2|2). In our paper, we describe the cup-product H1∨H1, where H1:=H1(osp(2|2),osp(1|2);Dλ,μ2) is the first differential osp(1|2)-relative cohomology of osp(2|2) with coefficients in Dλ,μ2 and Dλ,μ2:=Homdiff(Fλ2,Fμ2) is the space of linear differential operators acting on weighted densities. This result allows us to classify the osp(1|2)-trivial deformations of the osp(2|2)-module structure on the spaces of symbols Sd2. More precisely, we compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action. Furthermore, we prove that any formal osp(1|2)-trivial deformations of osp(2|2)-modules of symbols is equivalent to its infinitisemal part. This work is the simplest generalization of a result by Laraiedh [17].


Introduction
A smooth vector field on ℝ is represented by the Lie algebra (ℝ).Considering the (ℝ)-action on  ∞ (ℝ)'s 1-parameter deformation: where  ,  ∈  ∞ (ℝ) and  ′ ∶=   .The (ℝ)-module structure on  ∞ (ℝ), which is given by   for a fixed , is denoted by   .Geometrically, is the space of polynomial weighted densities of weight  ∈ ℝ.Along with the natural (ℝ)-action represented by  ,  (), D , ∶= Hom dif f (  ,   ) represents the (ℝ)-module of linear differential operators.The order of differential operators naturally filters each module D , , and the space of symbols is represented by the graded module  , ∶= grD , .The quotient-module D  , ∕D −1 , is isomorphic to  −− ; the principal "primary" symbol map  pr which given by: provides the isomorphism (see, e.g., [16]).As a result, the space  , can be expressed as   since it is an (ℝ)-module that depends exclusively on the difference  =  − , and we have as (ℝ)-modules.In the sense of Richardson-Neijenhuis [18], the space D , is a deformation of this space  , and it is not an isometric as a (ℝ)-module to the symbol space to which it corresponds.
Deformations of several categories of structures have played an increasingly important role in mathematics and a gain in physics during the last two decades.The objective of each of these deformation problems is to identify when all associated deformation obstructions disappear, and numerous attractive methods have been developed to determine when this occurs.The deformation theory of Lie algebras has received considerable attention.
The symmetry Lie algebra of the quantum system is typically an extension of the classical symmetry algebra (see [8]).As a result, central extensions are necessary in physics.
Richardson Neijenhuis [18,20] first considered some general questions about the theory in 1967.Deformations of Lie superalgebras considered by Binegar [9].Fialowski [13] extended the concept of deformations in 1988 by introducing deformations based on a complete algebra of a unique maximal ideal.Additionally and in general the concept of formal versal (or miniversal) deformation was introduced, and it was demonstrated that a versal deformation existed under certain cohomology constraints.Using this framework, Fialowski and Fuchs constructed a versal deformation [14].
Nijenhuis-Richardson asserts that the module deformation theory is inextricably linked to the determines the cohomology space.
For this statement to be more specific, being presented a Lie (super)-algebra ℭ, ℭ-module  and a  subalgebra of ℭ, then the cohomology space -relative H 1 (ℭ, ; End( )) measure infinitesimal trivial deformations in which the action is restricted to  (trivial deformations), where as the impediments to extending every infinitesimal -trivial deformation to a formal deformation are connected to H 2 (ℭ, ; End( )).
This main result has been implemented by numerous authors: In 1999, Ovsienko and Roger [19] classified the deformation of the Lie algebra of vector fields in side the Lie algebra of operators pseudodifferential on  1 .In 2002, Agrebaoui, Ammar, Lecomte and Ovsienko [1] classified the deformations multiparameter of the module of symbols of operators differential.In 2003, Agrebaoui, Nizar, Mabrouk and Ovsienko [2] classified the deformations of modules of differential forms.In 2008, Ben Ammar and Boujelbene [10] classified the deformations of   (ℝ)-modules of symbols trivial on (2).They are proved that the conditions of integrability of the infinitesimal deformation trivial on (2) are necessary and sufficient.Moreover, every formal deformations of   (ℝ)-modules of symbols trivial on (2) is equivalent to a polynomial one of degree ≤ 2. In 2009, Basdouri and al. [6] classified the deformations of (1)-modules of symbols trivial on (1|2).They are proved that the conditions of integrability of the infinitesimal deformation trivial on ℝ 1|1 are necessary and sufficient.Moreover, every formal deformations of (1)-modules of symbols is equivalent to a polynomial one of degree < 5. On the other hand, Ammar and Kammoun [4] classified the deformations of (1)-modules of symbols.
In 2010, Basdouri, Mabrouk, Bachir and Salem [7] classified the deformation of (1)-modules of symbols.They are proved that the conditions of integrability of the infinitesimal deformation of the second order are necessary and sufficient.In 2012, Basdouri and Ben Ammar [5] classified the deformation of (2)-modules and (1|2)-modules of symbols.In 2018, Ben Fraj, Abdaoui and Raouafi [11] classified the deformations of (2)-modules of symbols trivial on (1|2).They are proved that the conditions of integrability of the infinitesimal deformation of the second order are necessary and sufficient.Lately, in 2019, Laraiedh [17] classified the (1|1)-trivial deformations of (2|1)-modules of weighted densities on the superspace ℝ 1|2 .In this paper, first, we describe the cup-product H 1 ∨ H 1 .Second, we classify the (1|2)-trivial deformations of the (2|2)module structure on the superspace which is super analogous to the space   .We show that any formal deformation is equivalent to its infinitesimal part.
The Lie superalgebra (1|2) is easily determined to be a subalgebra of (2|2): The space of -densities is defined by: A differential operator on ℝ 1|2 is an operator with the following form on  ∞ (ℝ 1|2 ): Of course any differential operator defines a linear mapping   2 ⟼ ()  2 from  2  to  2  for any ,  ∈ ℝ, thus the space of differential operators becomes a family of (2)-modules  2 , for the natural action , can be expressed in this manner where   (,  1 ,  2 ) are arbitrary functions.
By definition, H  (ℭ, ; ) is the quotient space where   (ℭ, ; ) is the kernel of   and that it is called the space of -relative -cocycles and   (ℭ, ; ) is the elements in the range of  −1 and that it is called the space of -relative -coboundaries.For  = 1 and for all  ∈  1 (ℭ, ; ), ), for any , ℎ ∈ ℭ.

𝜷
We study the (1|2)-trivial deformations of the (2|2)-module structure on the space of symbols: The infinitesimal deformations are classified by: ) .
The space is spanned by: ) .
The space is spanned by: The space is spanned by: In our study, any infinitesimal (1|2)-trivial deformations of the (2|2)-module structure on  2  is of the form: where where the map: and the higher order terms  2  , The homomorphism condition (4.3) can be written as follows: [(), where the differential of the chain  is represented by  and for each linear map ,  ∶ ℭ ⟶ End( ), where ℭ is a Lie superalgebra and  is a vector superspace, ∨ represents the standard cup-product as described by: From (4.4) for any   , we can derive the following equation: The first obstruction to the integration of an infinitesimal deformation is presented by the first relation: Hence,  1 ∨  1 must therefore be a coboundary.
It is evident that the bilinear map  1 ∨  2 is an -relative 2-cocycle for any two 1-cocycles  1 and  2 ∈  1 (ℭ, ; End( )).Furthermore,  1 ∨  2 is a -relative 2-coboundary " 1 ∨  2 = " if one of the cocycles,  1 or  2 , is a -relative coboundary.As a result that the operation (4.4) generates a bilinear map: All the obstructions lie in H 2 (ℭ, ; End( )) and they are in the image of H 1 (ℭ, ; End( )) under the cup-product.Therefore, the cup-product H 1 ∨ H 1 is described in the following section.

The cup-product 𝐇 𝟏 ∨ 𝐇 𝟏
We have to distinguish two cases: Case 1: 2 ∉ ℕ.In this case, we have  (1) = Proof.In this case, the space H 1 ∨ H 1 is generated by the cup-product: , has a solution if and only if  = 0.
First of all, we have A.A. Almoneef, M. Abdaoui and A. Ghallabi For  = (, , ), we denote by   =      1   2 .Then, using equation (5.6), we can write ( Now, using the terms in ℎ in (5.6) for (, ) = ( 2 ,  2 ) then for (, ) = ( 2 ,  2 ), we get Similarly, the terms in  1 ℎ for (, ) = ( 1 ,  1 ) then for (, ) = ( 1 ,  1 ), we get Then, we have the following system: Proof.In this case, the space H 1 ∨ H 1 is generated by the two cup-products: Using simple computations, we confirm that of the natural action  of (2|2) on the space End( 2  ) and we investigate the necessary and sufficient conditions to extend it to a formal one: and the terms  2  ,   In addition, any formal deformation is equivalent to its infinitesimal part.
Proof.For the second-order terms, the condition (4.3) yields the following equation: = 0 for all  ≥ 0. Thus, the condition (6.9) is necessary.
We will explicitly find a deformation of   whenever conditions (6.9) are satisfied to demonstrate that conditions (6.9) are sufficient.It is possible to select zero for solution  (2) of (6.10).It is evident that a deformation (which is of order 1 in t) is obtained by selecting the highest-order terms  () with  ≥ 3, which are also identically zero.
The solutions   of the Maurer-Cartan equations (4.5) are defined up to a 1-cocycle and it has been proved in works [1,14] that different choices of solutions correspond to equivalent deformations.As a result we can always reduce   , for  = 2 to zero using equivalence.The highest-order terms   with  ≥ 3, then satisfy the equation (  ) and can be reduced to the identically zero map, via recurrence.□ Case 2: 2 =  ∈ ℕ.In this case, we have  (1) = ∑ Moreover, any formal deformation is equivalent to its infinitesimal part.
These factors provide a classification of the formal deformations, similar to the first case.

Now, for the
last system, we substituting the first equation into the second equation, adding the third and the fourth equation, the fifth and the sixth equation, substituting the seventh equation into the eighth equation and adding the ninth equation and the last equation, we yeilds the result  = 0. Therefore, we obtain the claim.□ Case 2: 2 =  ∈ ℕ.In this case, we have